1. Field of the Invention
This invention relates in general to the field of scanning interferometry and, in particular, to a new technique for improving the accuracy of iterative algorithms for interferometric measurements carried out on an interference microscope with modulation variations of the interference signal due to narrowband wavelengths and/or relatively high numerical apertures in addition to the presence of vibrations.
2. Description of the Prior Art
Many algorithms have been developed in the art for calculating surface topography from optical interference data recovered from conventional scanning techniques. In particular, phase-shifting interferometry (PSI) and related techniques are based on changing the phase difference between two coherent interfering beams using a single wavelength λ (ideally) and an optical system with zero numerical-aperture in some known manner, for example by changing the optical path difference (OPD) either continuously or discretely with time. Several measurements of light intensity with different OPD values, usually equally spaced, at a pixel of a photodetector can be used to determine the phase difference between the interfering beams at the point on a test surface corresponding to that pixel. Based on such measurements at all pixels with coordinates (x,y), a phase map Φ(x,y) of the test surface can be obtained, from which very accurate data about the surface profile may be calculated using well known algorithms. (For convenience, the term “pixel” is used hereinafter to refer both to a detector pixel and to the corresponding region of the sample surface. Also, the term “narrowband” is used exclusively to refer to a light with a spectral bandwidth, as opposed to a theoretically precise, single-wavelength source.)
Several factors, all well understood in the art, affect the quality of PSI interferometric measurements. For example, a fixed scanning step size between acquisition frames is generally assumed for all algorithms, but any factor that causes a change in step size (such as equipment vibrations, scanner nonlinearities, air turbulence, or wavelength variations in the case of wavelength scanning) can affect the performance of the algorithm and produce a non-uniform profile even when the sample surface is perfectly flat. Single-wavelength illumination and zero numerical aperture are also assumed for all algorithms, but not actually used in practice.
PSI corrects fairly well for miscalibrations and slow changes in step size, but it is ineffective for random changes in step size. Thus, algorithms based on an iterative approach have been developed to ameliorate this problem. See, for example, G. Lai et al., “Generalized Phase-Shifting Interferometry,” J. Opt. Soc Am. A, Vol. 8, No. 5, May 1991, p. 822; K. Okada et al., “Simultaneous Calculation of Phase Distribution and Scanning Phase Shift in Phase Shifting Interferometry,” Optics Communications, Vol. 84, Nos. 3-4, July 1991, p. 118; C. Wei et al. “General Phase-Stepping Algorithm with Automatic Calibration of Phase Step,” Opt. Eng. 38(8), August 1999, pp. 1357-1360; R. Onodera et al., “Phase-Extraction Analysis of Laser-Diode Phase-Shifting Interferometry that is Insensitive to Changes in Laser Power,” J. Opt. Soc Am. A, Vol. 13, No. 1, January 1996, p. 139; X. Chen et al., “Phase-Shifting Interferometry with Uncalibrated Phase Shifts,” Applied Optics, Vol. 39, No. 4, February 2000, p. 585; H. Guo et al., “Least-Squares Algorithm for Phase-Stepping Interferometry with an Unknown Relative Step,” Applied Optics, Vol. 44, No. 23, August 2005, p. 4854; H. Y. Yun et al., “Interframe Intensity Correlation Matrix for Self-Calibration in Phase-Shifting Interferometry,” Applied Optics, Vol. 44, No. 23, August 2005, p. 4860; I. Kong et al., “General Algorithm of Phase-Shifting Interferometry by Iterative Least-Squares Fitting,” Optical Engineering, Vol. 34, No. 1, January 1995, p. 183. In essence, all of these techniques involve a process whereby the result produced by the algorithm is refined by iteratively calculating improved step sizes, and correspondingly improved phases, that conform to the actual modulation data produced by the interferometric measurement. These iterative techniques typically find the phase and step size values that minimize an error function based on the difference between measured and theoretical intensity, the latter being calculated on the basis of equations with parameters expressed in function of x,y (sample pixel position) and z (vertical scan position).
All algorithms assume that the amplitude of modulation remains constant during the scan, but in fact that is almost never the case. In practice, the light intensity detected as a result of interference of the test and reference beams, which would be perfectly sinusoidal under ideal single-wavelength and zero-numerical-aperture conditions, exhibits a modulation variation that affects the interferometric result. FIGS. 1A and 1B illustrate this undesirable condition as might be seen at each pixel of the interferometer's detector. Thus, the effectiveness of the algorithm tends to decrease with increased spectral bandwidth of the source and also with increased numerical aperture of the objective used for the measurement.
In practice, a narrowband source (such as a laser) or a filtered broadband light is used instead of an ideal single-wavelength source to carry out the interferometric measurement, thereby affecting the amplitude of modulation of the interference signal and the performance of the algorithm. All systems also utilize a non-zero numerical aperture, which similarly affects the amplitude of modulation of the interference signal. Therefore, the effectiveness of all prior art algorithms tends to be undermined by the practical conditions under which they are normally implemented. PSI algorithms, which are designed to tolerate fairly well miscalibrated or slow changes in step size, tend to tolerate fairly well also variations of modulation in the interference signal caused by bandwidth and numerical aperture. However, they are very intolerant of random steps in the optical path of the interferometer due to mechanical vibration. (Mechanical vibrations are especially common when interference microscopes are employed in manufacturing environments.) The iterative techniques, on the other hand, are tolerant of random steps, but are very intolerant of variations in modulation of the interference signal.
In practice, all iterative algorithms used in the art neglect the effect that numerical apertures (in the order of 0.13 and greater) combined with narrowband wavelengths (in the order of 5 nm and greater, rather than single) have on the amplitude of the interference signal. Theoretically, an iterative procedure applied to a system with zero numerical aperture could produce reliable results with illumination of up to about 5-nm spectral bandwidth, and a system with single wavelength could produce reliable results with numerical apertures up to about 0.33. However, practical combinations of bandwidth and numerical aperture will produce a significant variation in the modulation amplitude of the interference signal. Thus, it would be very desirable to have an interferometric processing algorithm that accounted for the bandwidth of the light source and the numerical aperture that are actually used in interferometry. The present invention is directed at improving the prior art by introducing new parameters to reflect the effects of bandwidth and non-zero numerical aperture in the theoretical equations used to process the interferometric data.
The following derivation illustrates in detail the general approach used in the art to formulate the theoretical relationship between phase, scanning position and the intensity measured by the detector at each pixel as a result of an interferometric measurement. In addition, according to the invention, the approach is expanded to account for the effects of narrowband light and non-zero numerical aperture. While believed to be novel, this derivation is presented in the background section of this disclosure, rather than in the detailed description, in order to provide a foundation upon which the prior art as well as the present invention may be described. Though one skilled in the art will readily understand that the derivation is not unique, it is representative of the general theoretical equations that can be developed for the relationship between light intensity, phase, scanning position, numerical aperture and bandwidth.
The following general equation is generally used to represent the relationship between the intensity I of a light source of wavelength λ0 measured at each pixel and the scanning position zk at each acquisition frame k:
                                          I            ⁡                          (                              x                ,                y                ,                                  z                  k                                ,                                  λ                  0                                            )                                =                      B            +                          M              ⁢                                                          ⁢                              cos                ⁡                                  (                                                                                    4                        ⁢                        π                                                                    λ                        0                                                              ⁢                                          (                                                                        h                          obj                          0                                                -                                                  h                          ref                          0                                                -                                                  z                          k                                                                    )                                                        )                                                                    ,                            (        1        )            where I is a function of surface position x,y, scanning position zk, and wavelength λ0; hobj0 and href0 are the height positions of the object and reference surfaces, respectively, at the beginning of the scan; and B and M are background and modulation amplitude parameters, functions of x, y, zk and λ0. Note that zk is also a measure of the optical path difference between the reference and test beams of the interferometer. Therefore, while zk is used for convenience throughout this disclosure, it is intended also to refer more generally to OPD in any interferometric system, such as ones where interference is produced without an actual scan of the object surface. For the sake of precision, note also that the term I of Equation 1 expresses irradiance, rather than intensity, but intensity is normally used in the art to refer to both; therefore, intensity will be used throughout the description of the invention. Assuming that a fixed, single wavelength λ0 is used to perform phase-shifting interferometry, Equation 1 may be simplified by removing the explicit dependence on wavelength, which yields
                              I          ⁡                      (                          x              ,              y              ,                              z                k                                      )                          =                              B            ⁡                          (                              x                ,                y                ,                                  z                  k                                            )                                +                                    M              ⁡                              (                                  x                  ,                  y                  ,                                      z                    k                                                  )                                      ⁢                          cos              ⁡                              (                                                                            4                      ⁢                      π                                                              λ                      0                                                        ⁢                                      (                                                                  h                        obj                        0                                            -                                              h                        ref                        0                                            -                                              z                        k                                                              )                                                  )                                                                        (        2        )            
In practice the wavelength of the illumination source is known to span over some narrow band centered around λ0, such as illustrated in FIG. 2. Assuming such an asymmetrical illuminating spectrum of bandwidth W spanning from (λ0−λM) to (λ0+λM) around the center wavelength λ0, the sum of the contribution from each wavelength equidistant from λ0 can be written (from Equation 2) as follows:
                              I          ⁡                      (                          x              ,              y              ,                              z                k                            ,              λ                        )                          =                  B          +                      M            ⁢                                                  ⁢                          cos              ⁡                              (                                                                            4                      ⁢                      π                                                                                      λ                        0                                            -                      λ                                                        ⁢                                      (                                                                  h                        obj                        0                                            -                                              h                        ref                        0                                            -                                              z                        k                                                              )                                                  )                                              +                      N            ⁢                                                  ⁢                          cos              ⁡                              (                                                                            4                      ⁢                      π                                                                                      λ                        0                                            +                      λ                                                        ⁢                                      (                                                                  h                        obj                        0                                            -                                              h                        ref                        0                                            -                                              z                        k                                                              )                                                  )                                                                        (        3        )            where λ is the distance of each wavelength from λ0 (i.e., referring to FIG. 2, the two wavelengths at λ0−λ and λ0+λ), B=B(x,y,zk,λ), M=M(x,y,zk,λ), N=N(x,y,zk,λ), |λ|≦λM, B is background, and M and N represent signal modulation for the wavelengths λ0−λ and λ0+λ on each side of the center wavelength λ0.
Equation 3 may be simplified by normalizing its terms by defining
      h    ≡          λ              λ        0              ,            h      M        ≡                  λ        M                    λ        0              ,      φ    ≡          φ      ⁡              (                  x          ,          y                )              ≡                            4          ⁢          π                          λ          0                    ⁢              (                              h            obj            0                    -                      h            ref            0                          )              ,            and      ⁢                          ⁢              t        k              ≡                            4          ⁢          π                          λ          0                    ⁢                        z          k                .            For the case where
      0    ≤          λ              λ        0              ≤                  λ        M                    λ        0              =            h      M        ≤    0.1  (such as when a 70 nm bandwidth filter is used centered on 700 nm—note that in the art filter bandwidth refers to its width around the center wavelength, i.e., λ0−λM), the equation can be further simplified by using the first order Taylor expansion of the quantities 1/(λ0−λ) and 1/(λ0+λ), that is
            1                        λ          0                -        λ              ≅                  1                  λ          0                    ⁢              (                  1          +                      λ                          λ              0                                      )              =                              1                      λ            0                          ⁢                  (                      1            +            h                    )                ⁢                  1                                    λ              0                        +            λ                              ≅                        1                      λ            0                          ⁢                  (                      1            -                          λ                              λ                0                                              )                      =                  1                  λ          0                    ⁢                        (                      1            -            h                    )                .            Straightforward substitution yields the equationI(x,y,tk,λ0,h)=B+M cos[(φ−tk)(1+h)]+N cos[(φ−tk)(1−h)],  (4)which can be further simplified, applying the general trigonometric identity cos(a±b)=cos(a)cos(b)∓sin(a)sin(b), as followsI(x,y,tk,λ0,h)=B−{(M−N) sin[h(φ−tk)]}sin(φ−tk)+{(M+N)cos[h(φ−tk)]}cos(φ−tk).  (5)Note that Equation 5 reflects an intensity dependency on x, y and zk (through tk), as well as on wavelength spectrum (through h).
Integration of Equation 5 over h from 0 to hM (i.e., integrating the intensity equation over all wavelengths within the spectrum W of FIG. 2), leads to the equation
                              I          ⁡                      (                          x              ,              y              ,                              t                k                            ,                              λ                0                                      )                          =                  B          +                                    [                                                h                  M                                ⁡                                  (                                                            M                      eff                                        -                                          N                      eff                                                        )                                            ]                        ⁢                                                            cos                  ⁡                                      [                                                                  h                        M                                            ⁡                                              (                                                  φ                          -                                                      t                            k                                                                          )                                                              ]                                                  -                1                                                              h                  M                                ⁡                                  (                                      φ                    -                                          t                      k                                                        )                                                      ⁢                                                  ⁢                          sin              (                                                          ⁢                              φ                -                                  t                  k                                            )                                ⁢                                          +                                    [                                                          ⁢                                                h                  M                                ⁢                                                                  (                                                                  ⁢                                                      M                    eff                                    +                                      N                    eff                                                  )                            ]                        ⁢                                                  ⁢                                          sin                ⁡                                  [                                                            h                      M                                        ⁡                                          (                                              φ                        -                                                  t                          k                                                                    )                                                        ]                                                                              h                  m                                ⁡                                  (                                      φ                    -                                          t                      k                                                        )                                                      ⁢                          cos              ⁡                              (                                  φ                  -                                      t                    k                                                  )                                                                        (        6        )            where B=B(x,y,tk) now denotes a constant of integration with respect to λ, and Meff=Meff(x,y,tk) and Neff=Neff(x,y,tk) are effective mean values of M(x,y,tk,λ) and N(x,y,tk,λ) respectively, over the entire λ domain. Further, for a given λ0, by substituting according to the following definitions,
                              P          ≡                      P            ⁡                          (                              x                ,                y                ,                                  t                  k                                ,                                  λ                  0                                            )                                ≡                                    [                                                h                  M                                ⁡                                  (                                                            M                      eff                                        -                                          N                      eff                                                        )                                            ]                        ⁢                                                            cos                  ⁡                                      [                                                                  h                        M                                            ⁡                                              (                                                  φ                          -                                                      t                            k                                                                          )                                                              ]                                                  -                1                                                              h                  M                                ⁡                                  (                                      φ                    -                                          t                      k                                                        )                                                                    ,        and                            (                  7          ⁢          a                )                                          Q          ≡                      Q            ⁡                          (                              x                ,                y                ,                                  t                  k                                ,                                  λ                  0                                            )                                ≡                                    [                                                h                  M                                ⁡                                  (                                                            M                      eff                                        +                                          N                      eff                                                        )                                            ]                        ⁢                                                            sin                  ⁡                                      [                                                                  h                        M                                            ⁡                                              (                                                  φ                          -                                                      t                            k                                                                          )                                                              ]                                                  -                1                                                              h                  M                                ⁡                                  (                                      φ                    -                                          t                      k                                                        )                                                                    ,                            (                  7          ⁢          b                )            it is possible to write Equation 6 in much simpler form asI(x,y,tk)=B+P sin(φ−tk)+Q cos(φ−tk).  (8)The two parameters P and Q reflect the general asymmetry in the spectrum of narrowband light. If the spectrum were symmetrical, a single parameter would suffice.
Finally, applying the same trigonometric identity mentioned above [i.e., cos(a±b)=cos(a)cos(b)∓sin(a)sin(b)], as well as sin(a±b)=sin(a)cos(b)∓cos(a)sin(b), Equation 8 can be written to yieldI(x,y,tk)=B+[Q cos(φ)+P sin(φ)]cos(tk)+[Q sin(φ)−P cos(φ)]sin(tk),  (9)thereby separating the dependence on φ from the dependence on tk. Of course, the coefficients “P” and “Q” will still be dependent on both φ and tk and this dependence will not be separable in general. Equation 9 is a theoretical expression of intensity measured at a given pixel x,y as a function of initial phase φ and scanning position tk, wherein the narrowband nature of the light source (rather than single wavelength) is accounted for by the parameters B, P, and Q. These parameters remain dependent on x,y, tk and λ0.
The overall effect of numerical aperture (NA=sin(ΘMax)≧0) on the measured intensity was estimated by A. Dubois et al. in “Phase Measurements with Wide-Aperture Interferometers,” Applied Optics, May 2000, page 2326. For cases where NA is small (i.e., cosΘMax≅1), these authors determined that the effect of numerical aperture on light intensity can be accounted for by a coefficient equal to sin(β*tk)/(β*tk), where β=[1−cos(ΘMax)]/2. Thus, Equation 9 can be further modified to account for numerical aperture as follows:
                                                        I              NA                        ⁡                          (                              x                ,                y                ,                                  t                  k                                            )                                =                                    {                              B                +                                                      [                                                                  Q                        ⁢                                                                                                  ⁢                                                  cos                          ⁡                                                      (                            φ                            )                                                                                              +                                              P                        ⁢                                                                                                  ⁢                                                  sin                          ⁡                                                      (                            φ                            )                                                                                                                ]                                    ⁢                                      cos                    ⁡                                          (                                              t                        k                                            )                                                                      +                                                      [                                                                  Q                        ⁢                                                                                                  ⁢                                                  sin                          ⁡                                                      (                            φ                            )                                                                                              -                                              P                        ⁢                                                                                                  ⁢                                                  cos                          ⁡                                                      (                            φ                            )                                                                                                                ]                                    ⁢                                      sin                    ⁡                                          (                                              t                        k                                            )                                                                                  }                        *                                          sin                ⁡                                  (                                      β                    *                                          t                      k                                                        )                                                            β                *                                  t                  k                                                                    ,                            (        10        )            wherein the star symbol is used to denote multiplication and INA is intensity corrected for the effect of an approximately paraxial aperture (the normal condition for interferometric measurements).
Equation 10 may be simplified by defining
                                                        B              NA                        ≡                          B              ⁢                                                sin                  ⁡                                      (                                          β                      *                                              t                        k                                                              )                                                                    β                  *                                      t                    k                                                                                =                                    B              NA                        ⁡                          (                              x                ,                y                ,                                  t                  k                                ,                                  λ                  0                                            )                                      ,                            (                  11          ⁢          a                )                                                                    Q              NA                        ≡                          Q              ⁢                                                sin                  ⁡                                      (                                          β                      *                                              t                        k                                                              )                                                                    β                  *                                      t                    k                                                                                =                                                    [                                                      h                    M                                    ⁡                                      (                                                                  M                        eff                                            +                                              N                        eff                                                              )                                                  ]                            ⁢                                                sin                  ⁡                                      [                                                                  h                        M                                            ⁡                                              (                                                  φ                          -                                                      t                            k                                                                          )                                                              ]                                                                                        h                    M                                    ⁡                                      (                                          φ                      -                                              t                        k                                                              )                                                              ⁢                                                sin                  ⁡                                      (                                          β                      *                                              t                        k                                                              )                                                                    β                  *                                      t                    k                                                                        =                                          Q                NA                            ⁡                              (                                  x                  ,                  y                  ,                                      t                    k                                    ,                                      λ                    0                                                  )                                                    ,        and                            (                  11          ⁢          b                )                                                                    P              NA                        ≡                          P              ⁢                                                sin                  ⁡                                      (                                          β                      *                                              t                        k                                                              )                                                                    β                  *                                      t                    k                                                                                =                                                    [                                                                  ⁢                                                      h                    M                                    ⁡                                      (                                                                  M                        eff                                            -                                              N                        eff                                                              )                                                  ]                            ⁢                                                          ⁢                                                                    cos                    ⁡                                          [                                                                        h                          M                                                ⁡                                                  (                                                      φ                            -                                                          t                              k                                                                                )                                                                    ]                                                        -                  1                                                                      h                    M                                    ⁡                                      (                                          φ                      -                                              t                        k                                                              )                                                              ⁢                                                          ⁢                                                sin                  ⁡                                      (                                          β                      *                                              t                        k                                                              )                                                                    β                  *                                      t                    k                                                                        =                                          P                NA                            ⁡                              (                                  x                  ,                  y                  ,                                      t                    k                                    ,                                      λ                    0                                                  )                                                    ,                            (                  11          ⁢          c                )            where the subscript NA is used to indicate the numerical-aperture dependence of the parameter. Substituting from these definitions, Equation 10 becomesINA(x,y,tk)=BNA+[QNAcos(φ)+PNAsin(φ)]cos(tk)+[QNAsin(φ)−PNAcos(φ)]sin(tk).  (12)Note that Equation 12 also expresses theoretical intensity from a narrow-band light centered around λ0 measured at a given pixel x,y as a function of phase φ and scanning position tk, but now both the narrowband nature of the light source and numerical aperture are accounted for through the parameters BNA, PNA and QNA. For a given λ0, these parameters still remain dependent on x,y and tk.
The equation generally used in the art to calculate intensity has the formI(x,y,tk)=A(x,y,tk)+V(x,y,tk)*cos(φ−tk),  (13)and the functionality of “A” and “V” has been assumed separable, that is, A(x,y,tk)=A1(x,y)A2(tk) and V(x,y,tk)=V1(x,y)V2(tk). If no symmetry around the center wavelength λ0 is assumed to be present in the spectrum of the light used to carry out the interferometric measurement, the term PNA in Equation 12 derived above is zero (because Meff=Neff) and the equation reduces to:INA(x,y,tk)=BNA+[QNAcos(φ)]cos(tk)+[QNAsin(φ)]sin(tk),  (14)Again, through trigonometric identity, this equation can be written asINA(x,y,tk)=BNA+QNAcos(φ−tk),  (15)which is identical in form to the conventional relation of Equation 13, except that the parameters BNA and QNA have been derived so as to implicitly reflect spectral and numerical-aperture functionality. For the general case, the dependencies of QNA on φ and tk will no be separable. Because of this identity of forms, Equations 12, 14 and 15 will be used herein for consistency of notation, instead of Equation 13, first to explain the conventional approach to correcting height for errors introduced by nonlinearities and environmental perturbations. Then, the same equations will be used to explain the present invention and its novel features that distinguish it from all techniques heretofore used in the art.
Thus, referring to Equation 14, the conventional prior-art approach would involve separating the dependence of the terms over x,y from the dependence over tk [that is, writing BNA(x,y,tk)=B1(x,y)B2(tk) and QNA(x,y,tk)=Q1(x,y)Q2(tk)], so that the equation could be written as:I(x,y,tk)=B1(x,y,)B2(tk)+Q1(x,y)Q2(tk)cos(φ)cos(tk)+Q1(x,y)Q2(tk)sin(φ)sin(tk).  (16)This relation can be simplified by defining the quantitiesti C=C(x,y)≡Q1(x,y)cos[φ(x,y)]S=S(x,y)≡Q1(x,y)sin[φ(x,y)]Cos(tk)=Cos[tk(tk)]≡Q2(tk)cos(tk)Sin(tk)=Sin[tk(tk)]≡Q2(tk)sin(tk)such that Equation 16 can be written asI(x,y,tk)=B1(x,y)B2(tk)+C(x,y)*Cos(tk)+S(x,y)*Sin(tk).  (17)
In this equation the explicit dependence on φ of Equation 16 is removed and retained only implicitly in the substituted parameters C and S. Equation 17 is then used globally (i.e., over x, y and tk) to find iteratively the best sets of parameters [B1, B2, C, S, Cos(tk), Sin(tk)], and correspondingly the best sets of step sizes and phases, that match the modulation data registered during the interferometric scan.
To that end, an error function T is defined as the difference between the recorded values of intensity and the values predicted by Equation 17. Using a conventional least-squares approach, the error function may be defined as:
                              T          ⁡                      (                          x              ,              y              ,                                                          ⁢                              t                k                                      )                          ≡                                  ⁢                              ∑            x                    ⁢                                    ∑              y                        ⁢                                          ∑                k                            ⁢                                                                    [                                                                  I                        ⁡                                                  (                                                      x                            ,                            y                            ,                                                          t                              k                                                                                )                                                                    -                                                                                                    B                            1                                                    ⁡                                                      (                                                          x                              ,                              y                                                        )                                                                          ⁢                                                                              B                            2                                                    ⁡                                                      (                                                          t                              k                                                        )                                                                                              -                                                                        C                          ⁡                                                      (                                                          x                              ,                              y                                                        )                                                                          *                                                                                                  ⁢                                                                                                  ⁢                                                  Cos                          ⁡                                                      (                                                          t                              k                                                        )                                                                                              -                                                                        S                          ⁡                                                      (                                                          x                              ,                              y                                                        )                                                                          *                                                  Sin                          ⁡                                                      (                                                          t                              k                                                        )                                                                                                                ]                                    2                                .                                                                        (        18        )            Using Equation 18, the set of coordinates [B1, C, S] and [B2, Cos(tk), Sin(tk)] that minimize the error function T is determined iteratively in conventional manner by the following procedure, which is well documented in the literature.
Step A. The procedure is started with a guess solution [B20, Cos(tk0), Sin(tk0)]. The best set [B1i, Ci, Si] is found that minimizes the error function T (where the superscript “i” is used to denote the iteration number and i=1 in the first instance). This may be done in conventional manner by equating partial derivatives of T to zero and solving for the respective parameters. That is, each of the following identities is solved for its respective unknown variable, B1i, Ci, and Si:
                                          ∂            T                                ∂                          B              1                                      =                                            ∂              T                                      ∂              C                                =                                                    ∂                T                                            ∂                S                                      =            0.                                              (        19        )            
Step B. Using the newly determined values for [B1i, Ci, Si], the error function T is updated. Then, a set [B2i, Cos(tki), Sin(tki)] that minimizes the error function is similarly determined. This may be done again by solving the partial derivative equations
                                          ∂            T                                ∂                          B              2                                      =                                            ∂              T                                      ∂                              Cos                ⁡                                  (                                      t                    k                                    )                                                              =                                                    ∂                T                                            ∂                                  Sin                  ⁡                                      (                                          t                      k                                        )                                                                        =            0.                                              (        20        )            
Step C. Next, the step size tki (i.e., the frame separation during the scan) is estimated from the usual relation
                                          t            k            i                    =                      arc            ⁢                                                  ⁢                          tan              ⁡                              (                                                                            Sin                      ⁡                                              (                                                  t                          k                                                )                                                              i                                                                              Cos                      ⁡                                              (                                                  t                          k                                                )                                                              i                                                  )                                                    ,                            (        21        )            and tki is compared to the value obtained in the previous step. Initially, it is compared to the scanner's design step size, which will also be the step size for the interferometric algorithm in use (typically π/2). If the difference is less than a predetermined limit, the iteration process is stopped. Otherwise, the process is repeated from Step A, wherein the values of [B2i+1, Cos(tki+1), Sin(tki+1)] are updated using the previous solution [B2i, Cos(tki), Sin(tki)].
Step D. When the iterative process is stopped as described above, the phase (and correspondingly the height) is calculated for each pixel x,y in conventional manner using the equation
                              φ          =                                    arc              ⁢                                                          ⁢                              tan                ⁡                                  [                                                            sin                      ⁢                                                                                          ⁢                      φ                                                              cos                      ⁢                                                                                          ⁢                      φ                                                        ]                                                      =                          arc              ⁢                                                          ⁢                              tan                ⁡                                  (                                                            S                      i                                                              C                      i                                                        )                                                                    ,                            (        22        )            where i corresponds to the last step of iteration.
The conventional technique described above produces good results for narrowband light that approaches single wavelength illumination and for objectives without low numerical aperture. As the spectral band of the light source and/or the numerical aperture increase, the performance of the techniques currently in use deteriorates rapidly and the results are no longer reliable. For example, spectral bands in the order of 5 nm or greater produce a significant amount of modulation variation in the intensity data, which in turn causes erroneous height calculations. (Note that modulation variation is also commonly referred to as amplitude modulated sinusoidal signal in optics and electrical engineering.) Similar results are produced when numerical apertures greater than about 0.33 are used. Therefore, any procedure that allowed correction when narrowband light and/or significant numerical aperture are used would constitute a very useful advance in the art.